Length functions on mapping class groups and simplicial volumes of mapping tori

Let $M$ be a closed orientable manifold. We introduce two numerical invariants, called filling volumes, on the mapping class group $\mathrm{MCG}(M)$ of $M$, which are defined in terms of filling norms on the space of singular boundaries on $M$, both with real and with integral coefficients. We show that filling volumes are length functions on $\mathrm{MCG}(M)$, we prove that the real filling volume of a mapping class $f$ is equal to the simplicial volume of the corresponding mapping torus $E_f$, while the integral filling volume of $f$ is not smaller than the stable integral simplicial volume of $E_f$. We discuss several vanishing and non-vanishing results for the filling volumes. As applications, we show that the hyperbolic volume of $3$-dimensional mapping tori is not subadditive with respect to their monodromy, and that the real and the integral filling norms on integral boundaries are often non-biLipschitz equivalent.Comment: 20 pages; minor revision according to referee's suggestions. To appear at Annales de l'Institut Fourie

Integral filling volume, complexity and integral simplicial volume of 3-dimensional mapping tori

We show that the integral filling volume of a Dehn twist $f$ on a closed oriented surface vanishes, i.e. that the integral simplicial volume of the mapping torus with monodromy $f^n$ grows sublinearly with respect to $n$. We deduce a complete characterization of mapping classes on surfaces with vanishing integral filling volume and, building on results by Purcell and Lackenby on the complexity of mapping tori, we show that, in dimension three, complexity and integral simplicial volume are not Lipschitz equivalent.Comment: 24 pages, 4 figure

o-Benzoquinone dioxime

The title compound, C6H6N2O2, was obtained as a product of an in vitro study of the metabolism of benzofuroxan. The molecule exhibits a amphi configuration of the oxime groups C=N—OH. One oxime group is involved in the formation of a strong intra­molecular O—H⋯N hydrogen bond, while another links mol­ecules into zigzag chains along the c axis via inter­molecular O—H⋯N hydrogen bonds

The action of mapping class groups on de Rham quasimorphisms

We study the action of the mapping class group on the subspace of de Rham classes in the degree-two bounded cohomology of a hyperbolic surface. In particular, we show that the only fixed nontrivial finite-dimensional subspace is the one generated by the Euler class. As a consequence, we get that the action of the mapping class group on the space of de Rham quasimorphisms has no fixed points.Comment: 21 pages, 4 figures. Comments are welcome

On the amorphous layer in bone mineral and biomimetic apatite: A combined small- and wide-angle X-ray scattering analysis

The occurrence of an amorphous calcium phosphate layer covering the crystalline apatite core has been suggested to be an intrinsic feature of both bone mineral and synthetic biomimetic analogs. However, an exahustive quantitative picture of the amorphous-crystalline relationship in these materials is still miss- ing. Here, we present a multiple scale modelling that combines small-angle X-ray scattering (SAXS) and synchrotron wide-angle X-ray total scattering (WAXTS) analyses to investigate the amorphous-crystalline spatial interplay in bone sample and biomimetic carbonated nano-apatites. SAXS analysis indicates the presence of a single morphology consisting of tiny nanoplates (NPLs) and provides a measure of their thickness (falling in the 3–5 nm range). WAXTS analysis was performed by developing atomistic models of apatite NPLs incorporating lattice strain, mostly attributed to the carbonate content, and calculating the X-ray patterns using the Debye Scattering Equation. Upon model optimization, the size and strain parameters of the crystalline platelets were derived and the amorphous component, co-existing with the crystalline one, separated and quantified (in the 23–33 wt% range). Notably, the thickness of the ap- atite core was found to exhibit nearly null (bone) or minor ( < 0.5 nm, biomimetic samples) deviations from that of the entire NPLs, suggesting that the amorphous material remains predominantly distributed along the lateral sides of the NPLs, in a core-crown-like arrangement. The lattice strain analysis indicates a significant stiffness along the c axis, which is comparable in bone and synthetic samples, and larger deformations in the other directions. Statement of Significance Current models of bone mineral and biomimetic nanoapatites suggest the occurrence of an amorphous layer covering the apatitic crystalline nanoplates in a core-shell arrangement. By combining X-ray scatter- ing techniques in the small and wide angle regions, we propose a joint atomic-to-nanometre scale mod- elling to investigate the amorphous-crystalline interplay within the nanoplates. Estimates are extracted for the thickness of the entire nanoplates and the crystalline core, together with the quantification of the amorphous fraction and apatite lattice strain. Based on the thickness matching, the location of the amorphous material mostly along the edges of the nanoplates is inferred, with a vanishing or very thin layer in the thickness direction, suggesting a core-crown-like arrangement, with possible implications on the mineral surface reactivity.Fondazione Cariplo 2016-0648FEDER/Ministerio de Ciencia, Innovación y Universidades-Agencia Estatal de Investigación (FEDER/MCIU/AEI) RYC-2016-21042 RTI-2018-095794-A-C22FEDER/MCIU/AEI JdC-201

3-Phenyl­sulfanyl-4-phenyl­sulfonyl-1,2,5-oxadiazole 2-oxide

In the title compound, C14H10N2O4S2,the furoxan heterocyclic ring and the two S atoms are almost co-planar, with a mean deviation of 0.036 Å. The bond lengths in the penta­gonal ring show electron delocalization and the furoxan N—O bond length is quite short [1.211 (3) Å]. The dihedral angles between the central ring and pendant phenyl rings are 78.05 (14) and 84.28 (2)°

4-({4-[Bis(2-cyano­eth­yl)amino]­phen­yl}diazen­yl)benzene­sulfonamide

In the title compound, C18H16N6O2S, which belongs to the family of azo dyes, the dihedral angle between the benzene rings is 26.16 (7)°. In the crystal, mol­ecules are joined by N—H⋯N and C—H⋯N hydrogen bonds into double chains parallel to the a axis

C–halogen…O supramolecular synthons:in situcryocrystallisation of 1,2-dihalotetrafluoroethane/HMPA adducts

The in situ cryocrystallisation technique has been used to obtain four adducts between hexamethylphosphortriamide and 1,2-dihalotetrafluoroethanes having iodine, bromine and chlorine as halogen-bonding donor atoms. These systems allowed for a precise comparison of different C-X···O synthons. The effectiveness and reliability of the pharmacologically important C-Cl···O synthons are proven. © 2013 Copyright Taylor and Francis Group, LLC

2,3,4,6-Tetra-O-acetyl-2-phthalimido-β-d-glucopyran­oside

In the crystal structure of the title compound, C24H27NO11, a substituted tetra­acetyl glucopyran­oside derivative, weak inter­molecular C—H⋯O hydrogen bonds link the mol­ecules into ribbons propagated in [010]. The d configuration has been attributed on the basis of the synthesis and the β anomer has been determined from the structure